The number zero is, as Frege understands it, the number of the concept not-self-identical. That is: it’s the extension of that second-level concept under which fall exactly those first-level concepts that are equinumerous with the concept not-self-identical. Since there are, as a matter of logic, no non-self-identical objects, a concept is equinumerous with the concept not-self-identical iff nothing falls under it. Consider now a statement of the form “There are zero Gs,” which Frege understands to mean “0 = the number of Gs.” Given Frege’s analysis of “0” and of “the number of Gs,” this statement means:
“The extension of that concept under which fall all and only those concepts equinumerous with the concept not self-identical = the extension of that concept under which fall all and only those concepts equinumerous with the concept G.”
And this statement is straightforwardly equivalent (via the principle of extensionality) with the statement:
“The concept not-self-identical is equinumerous with the concept G,”
which itself is logically equivalent with the statement:
“∀x~Gx”
It’s worth pausing to notice just how important this point is for Frege. On the analysis in question, a statement about a particular object, 0, and its relation to a concept G, is logically equivalent to a claim about how many things fall under G. This is exactly what Frege needs in order to reconcile his two fundamental views about cardinal numbers: namely, that a statement of number is a statement about the cardinality of a concept, and that a statement of number is a statement about a particular object, that number. At least, this is what he needs for this reconciliation in the case of the number zero. In order to carry out this analysis for the rest of the finite cardinal numbers, Frege needs to provide canonical concepts guaranteed (via principles of pure logic) to have exactly the right number of objects falling under them. This is done as follows: the number one is, as Frege understands it, the number that belongs to the concept identical with zero. The number two is the number that belongs to the concept identical with zero or identical with one. And so on. In this way, Frege demonstrates how we can so understand the numbers that they are individual objects, and objects whose relations with arbitrary concepts “encode,” as it were, facts about the cardinalities of those concepts.
Having provided accounts of the individual numbers, Frege next sketches proofs, from (what he takes to be) purely logical principles, of a core of fundamental claims about number (essentially, variants of what we now know as the Dedekind-Peano axioms). The underlying idea was that from these fundamental claims we would be able in a routine way to prove the whole of the arithmetic of the finite cardinals, so that the proofs of these fundamental claims, once the details were filled in, would provide the decisive demonstration of the logicist thesis. Frege promises that the final demonstration will be provided in future work, work that will give the detailed and rigorous proofs of those fundamental arithmetical truths.
The semantic theory found in Grundlagen is similar to that of Begriffsschrift: Frege does not yet have the distinction between sense and reference, but does clearly hold that the important properties and relations in which he’s interested—that is, truth, logical entailment, and relations inter-definable with these—apply in the first instance not to sentences, but to the contents thereof. One important thesis insisted upon in Grundlagen, and one that proves highly influential throughout the semantic tradition that follows Frege, is what has come to be called the context principle. The principle is the thesis that in order to understand what the meaning of a word is, we should not take the word in isolation, but should consider the contribution made by that word to the sentences in which it appears. Frege takes it that this thesis is especially important when we investigate the meanings of numerals. The idea is that instead of asking, in isolation, what the numeral “3” stands for, that is, what the number three is, we should ask how that numeral contributes to the meanings of whole sentences in which it occurs. As Frege sees it, the failure to appreciate this point has been responsible for the mistaken idea that numerals refer to ideas, or to mental constructions of some kind: when people have looked for an object that can be identified in isolation, the only candidates have seemed to be some such psychological objects. Frege’s view is that the mistake lies in the initial focus on a search for objects that are identifiable independently of context (e.g. specific ideas), rather than on an investigation into the contribution made by numerals to whole statements. Adherence to the context principle, says Frege, enables us to make sense of a kind of object that’s identifiable in a quite different way than are, say, the objects of the material world. As we might put it, the context principle is essential to Frege’s account of our apprehension of abstract objects.
During the next decade, a good deal of Frege’s attention is presumably taken up with his monumental Grundgesetze der Arithmetik,5 whose first volume was published in 1893. This is the work in which Frege provides the remarkably rigorous proofs, promised in Grundlagen, of the fundamental truths about numbers. Prior to the publication of Grundgesetze, however, Frege writes and publishes three articles that have become enormously influential in the philosophy of language. These are “Über Sinn und Bedeutung” (“On Sense and Reference”),6 “Funktion und Begriff,”7 and “Über Begriff und Gegenstand.”8 In these essays, Frege introduces the distinction between sense and reference discussed earlier. Here we turn to some of the details of that mature semantic theory.
11.2.2 The 1891/92 Essays
Frege says very little about why, in the course of working out the very demanding proofs that were to establish his logicist thesis, he pauses to develop an account of the nature of language. Part of the explanation would seem, however, to be as follows. Frege’s view of arithmetic is that it is an objective science, in the sense that its truths hold independently of human activities. On this view, the fact that 13 is prime, for example, is in no way dependent on any ideas that anyone has ever had about 13; this thought would have been true whether anyone had ever entertained it or not. This means that the objects that arithmetic deals with, principally including the numbers, cannot be, and cannot be in any way, dependent on ideas or other human productions. Consider now the question of what we’re trying to prove when we prove that:
It would seem quite evident that what we’re trying to do here is quite different from what we’re trying to do when we attempt to prove that:
This fact, that to prove (D) is not the same thing as to prove (E), might be taken to provide an argument against Frege’s objectivist conception of arithmetic. The argument is as follows:
If arithmetic is about independently-existing objects, then, because the object 2+2 is the same as the object 4, (D) says exactly what (E) says, and hence a proof of (D) can be given just by proving the trivial (E). But this is absurd. So arithmetic is not about independently-existing objects.
The lesson to be drawn from the difference between (D) and (E), according to this line of thought, would be that (D) is about something like a “way of constructing” a number, or a pair of “ideas,” or some such thing. Because it’s plausible to claim that the idea or the “way of constructing” a number given by the phrase “2 + 2” is different from the idea or construction given by “4,” one can make sense, on this line of thought, of the clear difference between (D) and (E).
For Frege, it is important to resist this line of argument, and to maintain that arithmetic is really about the independently-existing arithmetical objects, despite the difference between, for example, (D) and (E). His response is that the difference between (D) and (E) is a matter of a difference in the senses, that is, the thoughts expressed by those sentences, despite the fact that their singular terms, that is, “2 + 2” and “4” have the same reference. The resulting two-tiered theory thus straightforwardly allows Frege to hold both that (D) and (E) express different items of knowledge (one of them being trivial, the other not), and that the sentences, and the thoughts themselves, are true in virtue
of purely objective facts about independently-existing objects.
On the mature semantic theory as it’s developed in this period, the thought expressed by a sentence is also known as the sense of that sentence, and is determined entirely by the senses of the sentence-parts and their order of composition. Thoughts are, as already discussed, the primary bearers of truth and falsehood, and of the logical relations. They are also the things with which one must be appropriately related in order to be said to understand the sentences that express them. When one is engaged in lucid conversation with someone, one grasps the thoughts expressed by the sentences used by that other person. This is done largely in virtue of one’s understanding of the other person’s words, that is, in virtue of one’s grasp of the senses associated with those words. Thoughts are also, finally, the contents of belief, knowledge, and the other attitudes. To believe or to doubt something is to bear a particular relation to a specific thought.
Thoughts therefore play a unifying role in the account of assertions, sentences, beliefs, and the truth-values thereof. The fact that one person can doubt just what another believes, and that this can be the content of another’s assertion or the claim expressed by a given sentence, is explained by Frege in terms of the fact that, in each case, the content in question is a thought. That these apparently-disparate entities—beliefs, assertions, sentences—can all be true or false is in turn explained by the fact that each is related in a straightforward way to a thought, and that this thought is, at bottom, the thing that is true or false.
The reference of a singular term (roughly, a noun phrase—e.g. a definite description, name, or pronoun) is an object (a person, place, physical object, number, etc.). The sense of a singular term is again what that term contributes to the senses of the sentences in which it appears, and is also, typically, a way of presenting the reference of that term. The other parts of language, for example, predicate phrases, relation terms, and so on, also have senses and references, according to Frege. The reference of a one-place predicate (e.g. “…is blue”) is what Frege now calls a concept. Concepts as conceived in this later period are similar to the concepts of Grundlagen in their close relationship to predicates, but they are essentially richer in two ways: First, they are clearly now the references of predicative phrases, and hence are not the entities grasped in an act of understanding. Secondly, concepts in the mature period are importantly different from objects in that they are essentially predicative. To understand this notion, consider the difference between the sentence (A) in section 11.1.3 and the list of terms
(AL) Alice, the property of liking geraniums
While (A) expresses a thought, (AL) doesn’t; the series of terms in the latter doesn’t form the kind of unified whole sufficient for expressing a truth or falsehood. Frege’s account of the difference is that the concept-phrase “…likes geraniums” in (A), unlike the phrase “the property of liking geraniums” in (AL), refers to something essentially predicative: the concept referred to is of such a nature that to refer to it is to predicate. Another way Frege has of putting the point is the metaphorical one that concepts are “unsaturated” or “incomplete,” unlike objects, which are “saturated” or “complete.” To use a concept-phrase in a sentence is to predicate, because concepts are just the kinds of things reference to which amounts to predication.
Concepts are, for Frege, a particular kind of function. Functions are unsaturated, that is, again, essentially predicative. Any part of a sentence that doesn’t refer to an object refers to a function. For example, in the sentence “2 + 2 = 4,” the part “…+…,” that is, the plus-sign accompanied by two gaps, refers to a function. Specifically, it is the function that takes as arguments pairs of numbers, and returns as value the sum of those numbers. Similarly, “the paternal grandfather of…” takes as argument a person, and returns as value the father of the father of that person. A concept (e.g. that referred to by “…likes geraniums” or “…is blue”) also gives values for arguments: in this case, the arguments are objects of appropriate kinds (something that does or doesn’t like geraniums, something that is or isn’t blue), while the value is what Frege calls a truth-value, that is, the value true or the value false. Concepts, in short, are one-place functions from objects to truth-values.9
In a complex term, like “Alice’s paternal grandfather,” the reference of the whole term is determined in the obvious way by function–argument application: the function referred to by “…’s paternal grandfather,” when applied to Alice, gives as value the person Robert (i.e. Alice’s paternal grandfather). Hence the reference of “Alice’s paternal grandfather” is simply the person Robert. (Notice that though we can now conclude that the two phrases “Alice’s paternal grandfather” and “Robert” have the same reference, they do not of course have the same sense.) Similarly, the reference of “the positive square root of nine” is the result of applying the function referred to by “the positive square root of…” to the argument nine, to yield three.
Frege’s view is that the truth-value of a sentence is determined in just the same way as is the reference of a complex singular term, via the application of function to argument. The truth-value of (A) is determined by applying the function referred to by “…likes geraniums” to the argument Alice, to deliver the value true. Because of the obvious parallels here, Frege uses the word “reference” to include not just the object referred to by a singular term and the function referred to by a function-expression, but also the truth-value of a sentence. Sentences, in short, are said to refer to truth-values.
Finally, function-expressions, including concept-expressions, have a sense as well as a reference. The senses are, predictably, simply the contributions made by those expressions to the thoughts expressed by the sentences that embed them. The phrases “…is a prime number between 12 and 16” and “…is a positive square root of 169,” though they refer to concepts that deliver the same value for every argument, nevertheless have different senses. This follows from the fact that a person might, for example, know that “There is exactly one positive square root of 169” is true while doubting whether “There is exactly one prime number between 12 and 16” is true.
To sum up Frege’s views about the senses and references of different parts of language: the reference of a sentence is a truth-value, while its sense is a thought. The reference of a singular term is an object, while its sense is the contribution made by that term to the thoughts expressed by sentences in which that term occurs. This sense is often usefully thought of as a “mode of presentation” of the object referred to. The reference of a function-expression is a function, an essentially predicative entity. The sense of such an expression is, again, just what that expression contributes to the thoughts expressed by sentences in which it appears.
There are also, as Frege sees it, terms that, though in some sense meaningful, lack reference. The phrase “the greatest prime number” is one such. Though the phrase clearly lacks a reference (since there’s no greatest prime), it is also not a nonsense phrase. We can, for example, prove rigorously that there is no greatest prime number, which we couldn’t do if the phrase in question were nonsense. Frege’s description of this situation is that the phrase “the greatest prime number” has a sense but no reference. Because the reference of a sentence is determined by function–argument application in the way described here, no sentence involving this term can have a reference either. That is to say, sentences like “The greatest prime number is even,” on Frege’s view, have no truth-value. They do, however, have a sense.
The essays published in 1891/92 also develop the application of the theory of sense and reference to various natural-language phenomena. The fundamental idea that the reference of a sentence is determined by the references of its parts, for example, is seen to face a potential difficulty when applied to sentences involving constructions like “Aristotle believed that…,” “Jürgen said that…,” and so forth. The difficulty is as follows: the ellipsis in these examples i
s typically filled by a complete sentence, resulting in a complex sentence like:
(S) Aristotle believed that humans are rational.
That embedded sentence (“humans are rational”) would normally, on Frege’s view, have a truth-value as its reference—in this case, the value true. But now something has gone wrong, since the truth-value of (S) is not determined just by the reference of “Aristotle believed that…” and the value true. We can see this by noticing that the sentence
(S*) Aristotle believed that the earth orbits the sun
has parts with the same ordinary references as do the parts of (S), but has a different truth-value. In short, the truth-value of the whole sentence, in the case of (S) and (S*), is not determined by what one would ordinarily take to be the references of its parts. Frege’s response to this difficulty is to say that in contexts involving the relation “…believed that…,” the sentence embedded to the right of that phrase has as its reference not its ordinary reference (i.e. its truth-value), but its ordinary sense (i.e. a thought). The sentence (S) is true iff Aristotle bears the relation of belief to the thought ordinarily expressed by “humans are mortal,” and accordingly, the phrase “humans are mortal,” in this context, refers to that thought. Similarly, in appropriate contexts, an embedded sentence might refer to itself (e.g. in quotation-mark contexts). In such cases, the embedded sentences have what Frege calls their “indirect” reference. The difficulty sketched here is thus averted, with the result that, uniformly, the reference of a sentence is determined by the references of its parts.
11.2.3 Mature Logicism: The Grundgesetze
In 1893, Frege publishes the first volume of his magnum opus, the Grundgesetze der Arithmetik. Much of what became the second volume was presumably completed by this time as well, though the second volume was not published until 190310. In Grundgesetze, Frege provides the rigorous development of arithmetic whose groundwork was laid in the Begriffsschrift of 1879 and the Grundlagen of 1884. Specifically, Grundgesetze introduces a formal system of logic very like that of Begriffsschrift, and demonstrates (i) how to analyze fundamental concepts and truths of arithmetic in terms of surprisingly-simple constituent concepts, and (ii) how to prove, with exacting rigor, the thus-analyzed arithmetical truths from (what he took to be) purely-logical principles, using purely-logical methods of inference. The analyses and proofs themselves exhibit an extraordinary degree of precision and rigor, reflecting Frege’s view that for his purposes, it is not enough that each step in a proof be obviously correct; it is required that the most fundamental principles on which each step is based be made evident to anyone who works step-by-step through the proofs. The proofs are, in this sense, a monument to analytic care and precision.
The Oxford Handbook of German Philosophy in the Nineteenth Century Page 42
